Name
Class
Date
Practice
5-5
Theorems About Roots of Polynomial Equations
Use the Rational Root Theorem to list all possible rational roots for each
equation. Then find any actual rational roots.
1. 36x + 144x  x  4 = 0
3
2
2. 12x + 14x  5x  14x  4 = 0
4
3
2
3. 5x  11x + 7x  1 = 0
3
2
A polynomial function P(x) with rational coefficients has the given roots. Find
two additional roots of P(x) = 0.
4. 2 + 3i and
7
6. 4i and 6  i
5. 3  2 and 1  3
7.
5 and  13
Write a polynomial function with rational coefficients so that P(x) = 0 has the
given roots.
8. 4 and 6
9. 3i and
10. 2 + i and 1  5
11.
6
3, 2,  i
What does Descartes’ Rule of Signs say about the number of positive real roots
and negative real roots for each polynomial function?
12. P(x) = 3x + x  8x  12
3
2
13. P(x) = 2x  x  3x + 7
4
3
14. P(x) = 4x  x  x + 6x  5
5
4
3
2
Find all rational roots for P(x) = 0.
15. P(x) = x + x + 10
16. P(x) = x + x  5x + 3
17. P(x) = 4x  12x  x + 3
18. P(x) = 12x + 31x  17x  6
3
3
2
3
2
3
2
19. Error Analysis A student claims that 2i is the only imaginary root of a polynomial
equation that has real coefficients. Explain the student’s mistake.
20. You are building a rectangular sandbox for a children’s playground. The width of the
sandbox is 4 times its height. The length of the sandbox is 8 ft more than 2 times its height.
You have 40 ft3 of sand available to fill this sandbox. What are the dimensions of the
sandbox?